For the next result, we introduce the standard parameterization of lattices in \(\mathbb {R}^2\) . Identifying \(\mathbb {R}^2\)  with \( via \) (x1,x2) x1 + i x2\(, we can view a lattice \) LR2\( as a subset of \) . It is a well-known fact (see e.g. [1]}) that any lattice \(\mathcal {L}\subset  can be given a basis \) r,r'\( such that the ratio \) = r'r\( belongs to the set{\begin{@align}{1}{-1} \lbrace \tau \in  \operatorname{Im}\tau > 0,\ |\tau | \ge 1,\ -\frac{1}{2} < \operatorname{Re}\tau \le \frac{1}{2} \rbrace , \end{@align}}which is the fundamental domain, \) H/SL(2, Z)\(, of the modular group \) SL(2, Z)\( acting on the Poincaré half-plane \) H\(.For a given \) L\(, the parameter \)\( is unique and is used as a parameterization (up to scaling) of the lattices. This gives the space of (normalized) lattices a topology.\)
